Hybrid system are of interest to a diverge group of researchers. Research topics include the mathematics and specification part, the verification and simulation implementation of these formal notions, and the physical semantics leg which tries to put the mathematical and implementation activity in a sound physical framework.

A mathematical paradigm for hybrid systems relies on the dynamic system notion of flows in phase space. An excellent book on dynamic systems is Differential Equations, Dynamical Systems, and Linear Algebra by Hirsh and Smale. The hybrid dynamic system paradigm is well phrased by Guckenheimer and Johnson in their "Planar Hybrid Systems" article in the Hybrid Systems III issue. Discrete transitions occur in between continuous modes of operation when state variables reach threshold values. These transitions may discontinuously change the field gradient that determines flow and may cause discontinuous jumps in the state vector values. Several implementations of this hybrid dynamic systems modeling paradigm have been developed, either from the discrete systems perspective or from the continuous systems point of view.

Discrete system behavior is well captured by finite state machines and these can be extended to include continuous behavior. Initially, this resulted in timed automata, which contain clocks that can be reset to 0 to constitute the continuous dimension. The resulting continuous behavior, therefore, is monotonously increasing with a constant rate and can be reset at discrete events. As a next step, hybrid automata were introduced that do not enforce the monotonicity constraint and allow clocks with different rates. These formalisms are especially useful in analyzing synchronization and communication behavior. For more information, refer to Alur, Courcoubetis, Henzinger, Halbwachs, Ho, Nicollin, Olivero, Sifakis, and Yovine, "The Algorithmic Analysis of Hybrid Systems" in Theoretical Computer Science.

In the continuous systems realm, there is a common mathematical framework in the form of differential equations, possibly extended by algebraic constraints (so called DAEs). State of the art modeling packages apply this formalism to handle large continuous models. To cope with the modeling complexity, hierarchy and object-oriented notions such as inheritance are applied. Presently, the international Modelica initiative aims to standardize the object-oriented model specification language. Models are specified as noncausal interactions between objects, and a compilation stage flattens the model structure and compiles a causal DAE. Noncausal modeling is first developed by the bond graph formalism and a good book on the subject is System Dynamics: A unified approach by Karnopp, Margolis, and Rosenberg. In the DAE framework, discrete mechanisms are introduced by conditionally (de)activating algebraic constraints, and, in some cases, some of the differential equations. This hybrid DAE formalism is powerful in modeling complex dynamic system behavior.

By nature, hybrid automata are well suited for verification purposes. To deal with complex continuous behavior, the continuous phase space is discretized into piecewise linear behaviors. In this context, linear refers to the constant rate of change constraint as opposed to the classical linear systems notion where dynamic behavior typically is exponential (i.e., linearity refers to the solutions not the underlying equations). An excellent book on systems theory is Introduction to Dynamic Systems: Theory, Models, and Applications by Luenberger. The piecewise linearization of a continuous phase space is studied in detail by Stursberg, Kowalewski, Hoffmann, and Preußig in their Hybrid Systems IV article "Comparing Timed and Hybrid Automata as Approximations of Continuous Systems".

Hybrid DAEs are a typical simulation formalism. Along with a continuous numerical approximation, root-finding to locate the discrete event time, event iteration to handle sequences of discrete events, and reinitialization needs to be handled. Cellier describes this in his thesis Combined Continuous/Discrete System Simulation by Use of Digital Computers: Techniques and Tools. Discontinuities that result from model abstraction (e.g., bouncing ball) can be conveniently modeled by hybrid DAEs as small, localized, conditional constraints. The elaborate discrete control mechanisms that are part of embedded control systems can be modeled by Petri nets, which requires a deterministic implementation.

The simulation and verification research takes a control perspective and relies on 'good models'. Such models are obtained by including physical constraints as developed in the field of thermodynamics. An excellent book (though in German) on the subject is Energie und Entropie: Eine Einfuhrung in die Thermodynamik by Falk and Ruppel. The resulting physical semantics for hybrid dynamic systems enables one to analyze mathematical singularities such as described by Utkin in Sliding Modes in Control and Optimization. Bond graphs apply thermodynamic concepts to provide a systematic modeling approach from physical concepts all the way to a mathematical representation that inherently enforces conservation of energy and continuity of power. Much work on including discontinuous phenomena in this theoretical modeling framework is done by Mosterman and Biswas, which resulted in Mosterman's dissertation Hybrid Dynamic Systems: A hybrid bond graph modeling paradigm and its application in diagnosis.

Finally, a lot of research is conducted to develop formal models for powerful hybrid system specification formalisms. Branicky presents a thorough study of this in his dissertation Studies in Hybrid Systems. This interesting strand of research has close ties with the formal methods community, which is an emerging factor in system specification for verification and, ultimately, automated code generation. This then addresses the eventual synthesis goal where an embedded control system is formulated as a system of equations with unknown control variables that are solved algorithmically.

In my opinion this describes the texture of the research performed by the hybrid dynamic systems community. A good indication of where we stand is the annual Hybrid System workshop of the IEEE Technical Committee on Hybrid Dynamical Systems. At present, we have moved beyond solving small, illustrative, problems and hybrid system methodologies are deployed in such projects as real-time hardware in the loop simulation and consumer electronics control verification. A suite of simulation and verification tools is readily available in the public domain.

Future research will apply hybrid dynamic system concepts and notions to increasingly complex systems for analysis and synthesis purposes based on sound theoretical principles. I look forward to new developments, enjoy being part of it, and hope this site contributes to advancing the state of the art!